Yolanda and Steve Turner with a picture of their late son Sean. Photograph: Tim Ireland/PA

The parents of a four-year-old boy who died following a heart operation are calling for an independent inquiry into the treatment of their son and that of other children at the same hospital after a coroner concluded there were "lost opportunities" in his care.

During an eight-day inquest into the death of Sean Turner, his parents, Yolanda and Steve, said they had been let down by a shortage of staff and a lack of expertise at Bristol Royal children's hospital.

They claimed that at one point Sean was so desperate for water while he recovered in ward 32, the children's cardiac ward, that he resorted to sucking liquid from moisturised tissues.

The Turners, from Warminster, Wiltshire, accused doctors of transferring their son from intensive care too soon and said staff there failed to pick up signs of his worsening condition.

Speaking at the end of the inquest on Thursday, they said: "There were many missed opportunities to rescue Sean from his desperate situation. In our opinion Sean was in the wrong hospital. We now have to try and rebuild our lives without our little boy."

The couple said the hospital had taken unacceptable risks. "Bristol had knowingly been carrying out the most complex surgery without sufficient high-dependency facilities and without a 24-hour integrated team," they said. "Although Sean needed a high level of nursing attention, at times on ward 32 he didn't even receive the most basic care. There was a lack of leadership, accountability and communication."

The hospital will face more questions in the coming months as four more inquests are held for children who were treated there. An inquest last year on another boy, seven-year-old Luke Jenkins, who died a month after Sean following treatment on ward 32, heard complaints strikingly similar to those of the Turners. Up to 10 families, including those of Sean and Luke, are taking or considering legal action against the hospital trust.

The Turners said they were concerned that patients at the hospital could still be at risk. "We have not seen enough evidence to persuade us that the lessons of Sean and Luke Jenkins' deaths, less than a month apart, have been learnt," they said. They are campaigning for an independent inquiry or for the parliamentary and health service ombudsman to step in.

Giving a narrative verdict, the Avon coroner, Maria Voisin, said Sean died in March 2012 from complications following an operation that took place six weeks earlier. She said: "There were lost opportunities to render medical care or treatment to Sean in this post-operative period."

The coroner said a follow-up operation could have been considered and more could have been done to tackle issues related to blood clotting. However, Voisin said she had not heard evidence of "gross failures to provide basic care" and said that having been told of changes the hospital had made she would not be writing a "prevention of future deaths" report to the trust.

She said: "I am aware that the trust has made lots of changes since Sean's death and I do not consider that I need to make any report in connection with this matter."

Robert Woolley, chief executive of the University Hospitals Bristol NHS foundation trust, apologised to the Turners. He said: "The inquest highlighted some missed opportunities in the care we gave to Sean when managing his post-operative complications and shortcomings in our communication with the family.

"I would like to offer my sincere apologies to Mr and Mrs Turner for the additional stress that we have caused them in relation to Sean's death. We are always improving our services and we have made significant changes since Sean was on the ward. Despite Sean's sad death, our outcomes are comparable to other national centres for this type of surgery."